Solving Inequalities: A Step-by-Step Guide

Hey math enthusiasts! Let's dive into the world of inequalities and figure out the correct representations for the given problem. We'll break down the steps, making it super easy to understand. So, grab your pencils, and let's get started!

Understanding the Inequality: $-3(2x - 5) < 5(2 - x)$

Inequalities are mathematical statements that compare two expressions, indicating that one is greater than, less than, greater than or equal to, or less than or equal to the other. In this case, we're dealing with the inequality $-3(2x - 5) < 5(2 - x)$. Our goal is to simplify this and find equivalent representations. The key to solving these kinds of problems is to apply the rules of algebra, keeping in mind that when we multiply or divide by a negative number, we must flip the inequality sign. Before we jump into the options, let's simplify the original inequality step-by-step. This initial simplification will help us determine which of the provided options are correct. By simplifying the original inequality, we can directly compare it with the given choices, making it easier to identify the correct representations. This process involves the application of the distributive property and combining like terms, which are fundamental concepts in algebra. This methodical approach ensures that we arrive at the correct answer efficiently and accurately. Always remember, precision is key in mathematics, and each step needs to be followed carefully.

First, we'll apply the distributive property to both sides of the inequality. The distributive property states that $a(b + c) = ab + ac$. Applying this to the left side, we get $-3 * 2x + (-3) * (-5)$, which simplifies to $-6x + 15$. On the right side, we have $5 * 2 + 5 * (-x)$, which simplifies to $10 - 5x$. So, our inequality now looks like this: $-6x + 15 < 10 - 5x$. This is our simplified form, and we'll use it to check the given options. Always double-check your work; a small mistake can lead to the wrong answer. This process of simplification is crucial to understand and master, as it forms the foundation for more complex algebraic manipulations. It's not just about getting the right answer; it's about understanding the underlying principles and the 'why' behind each step. Now, let's move on to the options and see which ones match our simplified inequality. Keep in mind that different representations can be correct as long as they are equivalent.

Examining the Options

Now, let's evaluate each option to see which correctly represents the original inequality. We'll carefully examine each option, comparing it to our simplified form and checking if any algebraic manipulations would make them equivalent. This systematic approach ensures that we thoroughly assess each choice, leaving no room for ambiguity. Remember, our goal is to identify options that are mathematically equivalent to the original inequality. This might involve applying the distributive property, combining like terms, or isolating the variable on one side. By taking it one step at a time, we can determine the validity of each representation and select the correct answers. We are looking for expressions that are equivalent to the original, which means they must have the same solution set. This requires a solid understanding of algebraic principles and the ability to apply them correctly. Let's see if we can find them!

Option A: $x < 5$

Option A states that $x < 5$. To determine if this is a correct representation, we need to solve our original inequality for $x$. We previously found that the simplified form of our original inequality is $-6x + 15 < 10 - 5x$. Let's start by adding $5x$ to both sides. This gives us: $-6x + 5x + 15 < 10 - 5x + 5x$, which simplifies to $-x + 15 < 10$. Next, subtract 15 from both sides: $-x + 15 - 15 < 10 - 15$. This simplifies to $-x < -5$. Finally, we need to divide both sides by $-1$. Remember, when you multiply or divide by a negative number, you must flip the inequality sign. Therefore, dividing both sides by $-1$ gives us $x > 5$. This result, $x > 5$, contradicts Option A, which states $x < 5$. Therefore, Option A is not a correct representation of the original inequality. Understanding how to isolate the variable and the importance of flipping the inequality sign is critical here. It's easy to overlook this step, but it's essential for arriving at the correct solution. Remember, the rules of algebra are your guide, and following them carefully ensures accuracy. Now let's move on to other options!

Option B: $-6x - 5 < 10 - x$

Let's analyze Option B, which is $-6x - 5 < 10 - x$. To see if this matches our original inequality, let's rearrange it and see if we can get it to match our simplified form. Our simplified form of the original inequality is $-6x + 15 < 10 - 5x$. Notice that in Option B, we have $-6x$ on the left side, which matches. However, it also includes $-5$ instead of $+15$. This suggests that Option B is derived from a different starting point or has an error in its algebraic manipulation. Let's work through the steps to see how we could solve this inequality and if it matches our original one. Let's add $x$ to both sides, which gives us $-5x - 5 < 10$. Then add $5$ to both sides, resulting in $-5x < 15$. Dividing both sides by $-5$, and remembering to flip the inequality sign (because we're dividing by a negative number), we get $x > -3$. This does not match our original inequality. Furthermore, this also does not match Option A. Therefore, we can confidently state that Option B is also incorrect, because it doesn't represent the original inequality correctly. A careful step-by-step approach always ensures we remain on the right track!

Option C: $-6x + 15 < 10 - 5x$

Let's evaluate Option C, which is $-6x + 15 < 10 - 5x$. Remember that in the first step of our solution, we already found that the simplified form of our original inequality is $-6x + 15 < 10 - 5x$. Comparing this with Option C, we see that it is an exact match! Option C is identical to the simplified form we derived from the original inequality. No further manipulation is needed; we can immediately see that this is a correct representation. This option correctly applies the distributive property and combines like terms. Therefore, we can conclude that Option C accurately represents the inequality. So, Option C is correct. The ability to recognize equivalent expressions is a key skill in algebra. It shows that you understand the underlying principles and can manipulate equations effectively. It also saves time during problem-solving. Make sure you can do this, it will help you in your future mathematics career!

Conclusion

Alright, guys! After carefully analyzing each option, we've found that Option C, $-6x + 15 < 10 - 5x$, is a correct representation of the inequality $-3(2x - 5) < 5(2 - x)$. Option A is incorrect because it simplifies to $x > 5$, and Option B is incorrect as it does not match the original. Understanding how to simplify and solve inequalities is a fundamental skill in algebra. The ability to manipulate and recognize equivalent expressions is critical. Keep practicing, and you'll become a pro in no time! Remember to always double-check your work and to pay close attention to the rules, especially when dealing with negative numbers. Keep up the excellent work, and always remember to enjoy the process of learning!